Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.
Citation: Qing-Hu Hou, Yarong Wei. Telescoping method, summation formulas, and inversion pairs[J]. Electronic Research Archive, 2021, 29(4): 2657-2671. doi: 10.3934/era.2021007
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Based on Gosper's algorithm, we present an approach to the telescoping of general sequences. Along this approach, we propose a summation formula and a bibasic extension of Ma's inversion formula. From the formulas, we are able to derive several hypergeometric and elliptic hypergeometric identities.
The telescoping method aims to solve the following problem: For a sequence
f(k)=g(k+1)−g(k). |
The sequence
b−1∑k=af(k)=g(b)−g(a). | (1) |
Gosper [7] solved the telescoping problem for hypergeometric terms by giving the so-called Gosper's algorithm. As mentioned in the book of Petkovše et al. [12], Gosper's algorithm is one of the landmarks in the history of symbolic summation. It not only fully solves the telescoping problem of hypergeometric terms, but also plays an important role in the creative telescoping algorithm developed by Wilf and Zeilberger [15].
Since the appearance of Gosper's algorithm and Zeilberger's algorithm, the telescoping method for more general sequences has been extensively studied. Chyzak [4] extended Zeilberger's algorithm to holonomic sequences. Inspired by Karr's summation algorithm [8], Schneider [13] presented an approach to summation in difference rings. Recently, Paule and Schneider [11] gave a symbolic summation theory for unspecified sequences.
This paper is motivated by considering the inversion formulas of basic and elliptic hypergeometric series. We find that the idea of Gosper's algorithm is an efficient mechanism for this purpose.
Let us recall the main steps of Gosper's algorithm. Suppose
tk+1tk=a(k)b(k)c(k+1)c(k), | (2) |
where
a(k)x(k+1)−b(k−1)x(k)=c(k). | (3) |
If
zk=b(k−1)x(k)c(k)tk. | (4) |
To deal with sequences which are not hypergeometric, we release the restriction that
This observation motivates a new approach to the definite summation of sequences beyond the hypergeometric ones. Given a sequence
Recall that two upper triangle matrices
n∑k=0fn,kgk,l=δn,l={1,if n=l,0,otherwise. | (5) |
If
n∑k=0fn,kF(k)=G(n)⟺n∑k=0gn,kG(k)=F(n). |
There are many interesting applications of the inversion pairs. For example, Warnaar [14] established some new identities on multibasic theta hypergeometric series by elliptic analogue of the inverse relations. Ma gave an extension of Warnaar's matrix inversion and obtained a series of classical identities in [10].
Ma [10] showed that the inversion relations (5) can be derived by summation formulas of form (1). This enables us to study inversion pairs by solving the telescoping problem. We will see that the trivial case of
Along this approach, we construct a new summation formula which extends Ma's result. Based on it and the related inversion pairs, we are able to derive several basic hypergeometric identities and elliptic hypergeometric identities.
The paper is organized as follows. In Section 2, we give a simple proof of Ma's summation formula based on the telescoping method. Then in Section 3, we propose a bibasic extension of Ma's summation formula along the telescoping approach and give some applications. Then in Section 4, we present a summation formula corresponding to the non trivial case where
Notation. Throughout this paper, we will employ the following standard notations for the theory of
(a;q)∞=∞∏k=0(1−aqk),and(a;q)n=(a;q)∞(aqn;q)∞. |
Write
(a1,a2,…,am;q)n=m∏j=1(aj;q)n, |
an basic hypergeometric series
rϕs[a1,a2,…,arb1,…,br;q,z]=∞∑k=0(a1,…,ar;q)k(q,b1…,bs;q)k[(−1)kq(k2)]1+s−rzk. |
and the balanced basic hypergeometric series
Assume further that
θ(x)=θ(x;p)=(x;p)∞(p/x;p)∞. |
The elliptic analogues of the
(a;q,p)∞=∞∏k=0θ(aqk;p),and(a;q,p)n=(a;q,p)∞(aqn;q,p)∞. |
Also, we write
(a1,a2,…,am;q,p)n=m∏j=1(aj;q,p)n,and(a;q,p)kn=(a,aq,…,aqk−1;qk,p)n. |
Note that
The balanced, very-well-poised, elliptic (or modular) hypergeometric series is defined by
r+1ωr(a1;a4,…,ar+1;q,p)=∞∑k=0θ(a1q2k;p)θ(a1;p)(a1,a4,…,ar+1;q,p)k(q,a1q/a4,…,a1q/ar+1;q,p)kqk, |
where the parameters satisfy
We follow the usual convention in defining produces as
m∏j=kAj:={AkAk+1⋯Am,m≥k,1,m=k−1,(Am+1Am+2⋯Ak−1)−1,m≤k−2. |
Ma used the following summation formula to prove the inversion formulas. We will give a simple proof by the idea of Gosper's algorithm.
Theorem 2.1. ([10,Theorem 4]). Let
f(a,c)g(b,d)−f(a,d)g(b,c)=f(a,b)g(c,d), | (6) |
and
m∑k=−nf(ak,bk)g(ck,dk)∏k−1j=1f(aj,cj)g(bj,dj)∏kj=1f(aj,dj)g(bj,cj)=∏mj=1f(aj,cj)g(bj,dj)∏mj=1f(aj,dj)g(bj,cj)−∏0j=−nf(aj,dj)g(bj,cj)∏0j=−nf(aj,cj)g(bj,dj). | (7) |
Proof. Let
tk=f(ak,bk)g(ck,dk)∏k−1j=1f(aj,cj)g(bj,dj)∏kj=1f(aj,dj)g(bj,cj). |
We have
tk+1tk=f(ak,ck)g(bk,dk)f(ak+1,dk+1)g(bk+1,ck+1)f(ak+1,bk+1)g(ck+1,dk+1)f(ak,bk)g(ck,dk). |
Set
a(k)=f(ak,ck)g(bk,dk), b(k−1)=f(ak,dk)g(bk,ck), c(k)=f(ak,bk)g(ck,dk). |
Then by the assumption (6), we have
a(k)−b(k−1)=c(k). |
Therefore, the telescoping of
zk=b(k−1)c(k)tk=∏k−1j=1f(aj,cj)g(bj,dj)∏k−1j=1f(aj,dj)g(bj,cj). |
we get the formula (7) by summing
Motivated by Ma's summation formula, we consider another special kind of
Theorem 3.1. Let
f(a,b)f(a,c)f(a,d)f(a,e)−f(b,1)f(c,1)f(d,1)f(e,1)=bf(a,1)f(a,bc)f(a,bd)f(a,be), | (8) |
and
m∑k=−nbkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek)×∏k−1j=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)∏kj=1f(bj,1)f(cj,1)f(dj,1)f(ej,1)=∏mj=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)∏mj=1f(bj,1)f(cj,1)f(dj,1)f(ej,1) |
−∏−n−1j=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)∏−n−1j=1f(bj,1)f(cj,1)f(dj,1)f(ej,1). | (9) |
Proof. Let
tk=bkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek)×∏k−1j=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)∏kj=1f(bj,1)f(cj,1)f(dj,1)f(ej,1). |
Then,
tk+1tk=bk+1f(ak+1,1)f(ak+1,bk+1ck+1)f(ak+1,bk+1dk+1)f(ak+1,bk+1ek+1)bkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek)×f(ak,bk)f(ak,ck)f(ak,dk)f(ak,ek)f(bk+1,1)f(ck+1,1)f(dk+1,1)f(ek+1,1). |
Set
a(k)=f(ak,bk)f(ak,ck)f(ak,dk)f(ak,ek),b(k)=f(bk+1,1)f(ck+1,1)f(dk+1,1)f(ek+1,1),c(k)=bkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek). |
We see that (2) holds. By the assumption (8), we have
a(k)−b(k−1)=c(k). |
Hence by the Gosper's algorithm, we obtain the telescoping of
zk=b(k−1)c(k)tk=∏k−1j=1f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)∏k−1j=1f(bj,1)f(cj,1)f(dj,1)f(ej,1). |
we get the formula (9) by summing
Now we give some applications of Theorem 3.1. The first one is a summation formula for finite series.
Corollary 3.1. Let
m∑k=−nbk(1−ak)(1−ak/bkck)(1−ak/bkdk)(1−ak/bkek)×∏k−1j=1(1−aj/bj)(1−aj/cj)(1−aj/dj)(1−aj/ej)∏kj=1(1−bj)(1−cj)(1−dj)(1−ej)=∏mj=1(1−aj/bj)(1−aj/cj)(1−aj/dj)(1−aj/ej)∏mj=1(1−bj)(1−cj)(1−dj)(1−ej)−∏−n−1j=1(1−aj/bj)(1−aj/cj)(1−aj/dj)(1−aj/ej)∏−n−1j=1(1−bj)(1−cj)(1−dj)(1−ej). | (10) |
Proof. Let
Corollary 3.1 is equivalent to a result of Ian G. Macdonald, first published by Bhatnagar and Milne [2,Eq.(2.30)]. From this formula, we can derive several summation identities. Here are two examples.
Example 3.1. In Corollary 3.1, set
ak=aq2k, bk=qk, ck=aqk/b, dk=aqk/c, ek=aqk/d, |
where
m∑k=0(a,qa1/2,−qa1/2,b,c,d;q)k(q,a1/2,−a1/2,aq/b,aq/c,aq/d;q)k(aqbcd)k=(aq,aq/bc,aq/bd,aq/cd;q)m(aq/b,aq/c,aq/d,aq/bcd;q)m. |
Let
_6ϕ5[a,qa1/2,−qa1/2,b,c,da1/2,−a1/2,aq/b,aq/c,aq/d;q,aqbcd]=(aq,aq/bc,aq/bd,aq/cd;q)∞(aq/b,aq/c,aq/d,aq/bcd;q)∞. |
Example 3.2. Setting
ak=adqkpk, bk=dqk, ck=adqk/b, dk=adpk/x, ek=bxpk/d |
in Corollary 3.1, we derive an indefinite bibasic summation formula [6,Eq. (Ⅱ.36)]:
m∑k=−n(1−adpkqk)(1−bpkq−k/d)(1−ad)(1−b/d)(a,b;p)k(x,ad2/bx;q)k(dq,adq/b;q)k(adp/x,bpx/d;p)kqk=(1−a)(1−b)(1−x)(1−ad2/bx)(1−ad)(1−b/d)(d−x)(1−ad/bx){(ap,bp;p)m(xq,ad2q/bx;q)m(dq,adq/b;q)m(adp/x,bpx/d;p)m−(1/d,b/ad;q)n+1(x/ad,d/bx;p)n+1(1/a,1/b;p)n+1(1/x,bx/ad2;q)n+1}. |
Gasper and Rahman used this identity to set up a series of quadratic and cubic summation and transformation formulas of basic hypergeometric series.
The second one is an elliptic hypergeometric identity.
Corollary 3.2. Let
m∑k=−nbkθ(ak,ak/bkck,ak/bkdk,ak/bkek;p)∏k−1j=1θ(aj/bj,aj/cj,aj/dj,aj/ej;p)∏kj=1θ(bj,cj,dj,ej;p)=∏mj=1θ(aj/bj,aj/cj,aj/dj,aj/ej;p)∏mj=1θ(bj,cj,dj,ej;p)−∏−n−1j=1θ(aj/bj,aj/cj,aj/dj,aj/ej;p)∏−n−1j=1θ(bj,cj,dj,ej;p). | (11) |
Proof. Let
θ(a/b,a/c,a/d,a/e;p)−θ(b,c,d,e;p)=bθ(a,a/bc,a/bd,a/be;p), |
which means that
This formula has been stated in an equivalent form by Warnaar [14,Eq.(3.2)]. We list some applications of Corollary 3.2 on elliptic hypergeometric series.
Example 3.3. Setting
ak=q2ka, bk=qk, ck=qka/b, dk=qka/c, ek=qka/d |
and
m+1∑k=0θ(q2ka;p)θ(a;p)(a,b,c,d,a2qm+2/bcd,q−m−1;q,p)k(q,qa/b,qa/c,qa/d,bcdq−m−1/a,aqm+2;q,p)kqk=(qa,qa/bc,qa/bd,qa/cd;q,p)m+1(qa/b,qa/c,qa/d,qa/bcd;q,p)m+1. |
Example 3.4. Setting
ak=aqkrk, bk=qk/bd, ck=adqk, dk=crk, ek=abrk/c |
and
m∑k=0qkθ(aqkrk,bq−krk;p)θ(a,b;p)(abd,1/d;r,p)k(cr,abr/c;r,p)k(a/c,c/b;q,p)k(q/bd,adq;q,p)k=θ(c,ab/c,bd,ad;p)θ(a,b,abd/c,cd;p)(1−(abd,dr−m;r,p)m+1(r−m/c,ab/c)m+1(a/c,bq−m/c;q,p)m+1(bdq−m,ad;q,p)m+1). |
It is an indefinite elliptic hypergeometric series of Warnaar [14]. Moreover, Warnaar used the specialization of this identity to set up a pair of inverse matrices.
Example 3.5. Setting
ak=ad(rst/q)k, bk=dqk, ck=adb(st/q)k, dk=adc(rt/q)k, ek=bcd(rs/q)k |
in Corollary 3.2, and making some simplifications, we deduce that [6,Page 326,Eq.(11.6.6)]
n∑k=−mθ(ad(rst/q)k,brk/dqk,csk/dqk,adtk/bcqk;p)θ(ad,b/d,c/d,ad/bc;p)×(a;rst/q2,p)k(b;r,p)k(c;s,p)k(ad2/bc;t,p)k(dq;q,p)k(adst/bq;st/q,p)k(adrt/cq;rt/q,p)k(bcrs/dq;rs/q,p)kqk=θ(a,b,c,ad2/bc;p)dθ(ad,b/d,c/d,ad/bc;p)×{(arst/q2;rst/q2,p)n(br;r,p)n(cs;s,p)n(ad2t/bc;t,p)n(dq;q,p)n(adst/bq;st/q,p)n(adrt/cq;rt/q,p)n(bcrs/dq;rs/q,p)n−(c/ad;rt/q,p)m+1(d/bc;rs/q,p)m+1(1/d;q,p)m+1(b/ad;st/q,p)m+1(1/c;s,p)m+1(bc/ad2;t,p)m+1(1/a;rst/q2,p)m+1(1/b;r,p)m+1}. |
In this section, we consider the case of nontrivial
Theorem 4.1. Let
x(k+1)x(k)=g(ak,pk)g(akbkck,pkqkrk)−g(akbk,pkqk)g(akck,pkrk)g(bk,qk)g(ck,rk). | (12) |
Then for any nonnegative integers
m∑k=−nx(k)g(akbk,pkqk)g(akck,pkrk)∏k−1j=1g(bj,qj)g(cj,rj)∏kj=1g(aj,pj)g(ajbjcj,pjqjrj)=x(−n)−n−1∏j=1g(bj,qj)g(cj,rj)g(aj,pj)g(ajbjcj,pjqjrj)−x(m+1)m∏j=1g(bj,qj)g(cj,rj)g(aj,pj)g(ajbjcj,pjqjrj). | (13) |
Proof. Let
tk=−x(k)g(akbk,pkqk)g(akck,pkrk)∏k−1j=1g(bj,qj)g(cj,rj)∏kj=1g(aj,pj)g(ajbjcj,pjqjrj). |
Then,
tk+1tk=−x(k+1)g(ak+1bk+1,pk+1qk+1)g(ak+1ck+1,pk+1rk+1)−x(k)g(akbk,pkqk)g(akck,pkrk)×g(bk,qk)g(ck,rk)g(ak+1,pk+1)g(ak+1bk+1ck+1,pk+1qk+1rk+1). |
Set
a(k)=g(bk,qk)g(ck,rk),b(k)=g(ak+1,pk+1)g(ak+1bk+1ck+1,pk+1qk+1rk+1),c(k)=−x(k)g(akbk,pkqk)g(akck,pkrk). |
We see that (2) holds. By the assumption (12), we have
a(k)x(k+1)−b(k−1)x(k)=c(k). |
Hence the telescoping of
zk=b(k−1)x(k)c(k)tk=x(k)k−1∏j=1g(bj,qj)g(cj,rj)g(aj,pj)g(ajbjcj,pjqjrj). |
We get the formula (13) by summing
Actually, given an arbitrarily function
To derive hypergeometric identities, we take
Corollary 4.1. Let
m∑k=−n(−1)k(akbk−pkqk)(akck−pkrk)∏k−1j=1ajpj(bj−qj)(cj−rj)∏kj=1(aj−pj)(ajbjcj−pjqjrj)=(−1)n−n−1∏j=1ajpj(bj−qj)(cj−rj)(aj−pj)(ajbjcj−pjqjrj)+(−1)mm∏j=1ajpj(bj−qj)(cj−rj)(aj−pj)(ajbjcj−pjqjrj). | (14) |
Proof. Let
x(k+1)x(k)=−akpk. |
Considering the initial value, we have
x(k)=(−1)kk−1∏i=0aipi. |
Hence by Theorem 4.1 we obtain (14).
As examples, we derive the following hypergeometric and basic hypergeometric identities.
Example 4.1. Setting
ai=i+1, bi=p+i+1, ci=p2+i+1, pi=1, qi=1, ri=1 |
in (14) where
x(k)=(−1)kk!. |
After some simplifications, we derive that
m∑k=0(−1)kk(p+k−1p)(p2+k−1p2)×((k+1)(p+k+1)−1)((k+1)(p2+k+1)−1)∏kj=1(j(p+j)(p2+j)−1)=(−1)m(p+mp)(p2+mp2)(m+1)∏mj=1(j(p+j)(p2+j)−1). |
Example 4.2. Setting
ai=(i+1)2, bi=(p+i+1)2, ci=(p2+i+1)2, pi=1, qi=1, ri=1 |
in (14), we see that
x(k)=(−1)k(k!)2. |
After some simplifications, we derive that
m∑k=0(−1)kk2(p+k−1p)(p2+k−1p2)(p+k+1p+2)(p2+k+1p2+2)×((k+1)2(p+k+1)2−1)((k+1)2(p2+k+1)2−1)∏kj=1(j2(p+j)2(p2+j)2−1)=(p+mp)(p2+mp2)(p+m+2p+2)(p2+m+2p2+2)(−1)m(m+1)2∏mj=1(j2(p+j)2(p2+j)2−1). |
Example 4.3. Let
x1=cri, x2=bqi, x3=api, x4=dsi |
Setting
ai=x1−x2, bi=x1−x3, ci=x2−x4, |
pi=1/x2−1/x1, qi=1/x3−1/x1, ri=1/x4−1/x2 |
in (14), we see that
x(k)=(b/c;q/r)k(bc;qr)kb2kq2(k2). |
After some simplifications, we derive that [1,Exapmle 6.2]
m∑k=0q(k+12)(1−abpkqk)(1−apk/bqk)(1−cdrksk)(1−dsk/crk)f(k)=(b−c)(1−1/bc)(a−d)(ad−1)+q(m2)(bdqmsm−1)(bqm−dsm)(crm−apm)(acpmrm−1)bcrmf(m), |
where
f(k)=(a/c;p/r)k(ac;pr)k(d/b;s/q)k(bd;qs)kbkakp(k+12)(bq/cr;q/r)k(bcqr;qr)k(ds/ap;s/p)k(adps;ps)k. |
Now we consider another choice of the function
Corollary 4.2. Let
m∑k=−n(−1)k(1−akbkpkqk)(1−akckpkrk)∏k−1j=1ajpj(1−bjqj)(1−cjrj)∏kj=1(1−ajpj)(1−ajbjcjpjqjrj)=(−1)n−n−1∏j=1ajpj(1−bjqj)(1−cjrj)(1−ajpj)(1−ajbjcjpjqjrj)+(−1)mm∏j=1ajpj(1−bjqj)(1−cjrj)(1−ajpj)(1−ajbjcjpjqjrj). | (15) |
Proof. Let
x(k+1)x(k)=−akpk. |
Considering the initial value, we derive that
x(k)=(−1)kk−1∏i=0aipi. |
Hence by Theorem 4.1 we obtain (15).
Example 4.4. Set
ak=a, bk=b, ck=c, pk=pk, qk=qk, rk=rk |
in (15), we see that
x(k)=(−a)kp(k2). |
After some simplifications, we derive
m∑k=0(−a)kp(k2)(1−abpkqk)(1−acpkrk)(b;q)k(c;r)k(ap;p)k(abcpqr;pqr)k=(a−1)(abc−1)+(−a)map(m+12)(b;q)m+1(c;r)m+1(ap;p)m(abcpqr;pqr)m. |
We remark that this formula is a result of Gosper, first published by Bauer and Petkovšek [1,Eq.(6.39)].
In this section, we will derive an inversion pair via the summation formula (9) and present some applications.
To construct the inversion pair, we first take
Lemma 5.1. Let
n∑k=0bkf(ak,1)f(ak,bkck)f(ak,bkdk)f(ak,bkek)×k−1∏j=0f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)n∏j=k+1f(bj,1)f(cj,1)f(dj,1)f(ej,1) |
=n∏j=0f(aj,bj)f(aj,cj)f(aj,dj)f(aj,ej)−n∏j=0f(bj,1)f(cj,1)f(dj,1)f(ej,1). | (16) |
Now we are ready to give the inversion pair.
Theorem 5.1. Let
f(x,y)=−(x/y)f(y,x),f(tx,ty)=f(x,y). | (17) |
for any variable
fn,k=f(z,z/xkyk)f(z2xk/yk,z)f(zxn,z/yn)f(z2xn/yn,z)∏nj=k+1f(zxn/yj,1)f(xnyj,1)∏n−1j=kxjf(xn,xj)f(zxn,1/xj) | (18) |
and
gn,k=∏n−1j=kf(xk,1/yj)f(zxk,yj)∏nj=k+1xjf(xk/xj,1)f(zxkxj,1). | (19) |
Then
Proof. Since
fn,n=f(z,z/xnyn)f(z2xn/yn,z)f(zxn,z/yn)f(z2xn/yn,z)=f(z,z/xnyn)f(zxn,z/yn)=1. |
Therefore it holds for
Now assume
aj=zxlxn, bj=zxn/yj+l, cj=xnyj+l, dj=xl/xj+l, ej=zxlxj+l |
in (16). The first product of the right hand side of (16) contains the term
f(an−l,en−l)=f(zxlxn,zxlxn) |
and the second product of the right hand side of (16) contains the term
f(d0,1)=f(xl/xl,1). |
Noting that
By the relation (17), the left hand side becomes
n∑k=lzxnykf(zxlxn,1)f(zxl,zxn)f(z,z/xkyk)f(z,z2xk/yk)×n∏j=k+1f(zxn/yj,1)f(xnyj,1)f(xl/xj,1)f(zxlxj,1)×k−1∏j=lf(xl,1/yj)f(zxl,yj)f(zxn,1/xj)f(xn,xj) |
=n∑k=l−xnxkf(zxlxn,1)f(zxl,zxn)f(z,z/xkyk)f(z2xk/yk,z)×n∏j=k+1f(zxn/yj,1)f(xnyj,1)f(xl/xj,1)f(zxlxj,1)×k−1∏j=lf(xl,1/yj)f(zxl,yj)f(zxn,1/xj)f(xn,xj). |
Dividing the factor
−xnf(zxlxn,1)f(zxl,zxn)f(zxn,z/yn)f(z2xn/yn,z)n∏j=lxj×n−1∏j=lf(zxn,1/xj)f(xn,xj)n∏j=l+1f(xl/xj,1)f(zxlxj,1), |
we could get it.
It is straightforward to check that
Corollary 5.1. Let
fn,k=(1−xkyk)(1−zxk/yk)(1−xnyn)(1−zxn/yn)∏nj=k+1(1−zxn/yj)(1−xnyj)∏n−1j=kxj(1−xn/xj)(1−zxnxj), |
and
gn,k=∏n−1j=k(1−xkyj)(1−zxk/yj)∏nj=k+1xj(1−xk/xj)(1−zxkxj). |
Then
We remark that this inversion pair was first given by Krattenthaler [9,Eq.(1.5)]. From this inversion pair, we can derive several classical inversion pairs.
Example 5.1. Set
fn,k=(1−aq2k)(b;q)n+k(ba−1;q)n−k(ba−1)k(1−a)(aq;q)n+k(q;q)n−k |
and
gn,k=(1−bq2k)(a;q)n+k(ab−1;q)n−k(ab−1)k(1−b)(bq;q)n+k(q;q)n−k. |
This pair is due to Bressoud [3], which was used to study the finite forms of Rogers-Ramanujan identities.
Example 5.2. Set
fn,k=pk−n(1−bpkqk)(1−ap−kqk/b)(bpnqn;p−1)n−k(bpnq−n/a;p−1)n−k(1−bpnqn)(1−ap−nqn/b)(aq2n−1;q−1)n−k(q−1;q−1)n−k |
and
gn,k=(bpkqk;q−1)n−k(bpkq−k;q−1)n−k(aq2k+1;q)n−k(q;q)n−k. |
Notice that it is an inversion pair that involves rising
By the definition of theta functions, we see that
Corollary 5.2. Let
fn,k=θ(xkyk;p)θ(zxk/yk;p)θ(xnyn;p)θ(zxn/yn;p)∏nj=k+1θ(zxn/yj;p)θ(zxnyj;p)∏n−1j=kxjθ(xn/xj;p)θ(zxnxj;p) |
and
gn,k=∏n−1j=kθ(xkyj)θ(zxk/yj)∏nk+1xjθ(xk/xj)θ(zxkxj). |
Then
This is a pair of inverse matrices due to Warnaar [14] and the elliptic analogue of inverse matrices of Krattenthaler [9,Eq.(1.5)]. Moreover, we can derive the classical pairs of inverse matrices.
Example 5.3. Set
fn,k=(−1)n−kq(n−k2)θ(aqkrk;p)θ(qkr−k/b;p)θ(aqnrn;p)θ(qnr−n/b;p)(aqk+1rn,qk+1r−n/b;q,p)n−k(r,abrn+k;r,p)n−k |
then
gn,k=(aqkrk,qkr−k/b;q,p)n−k(r,abr2k+1;r,p))n−k |
We note that the case of
Example 5.4. Set
fn,k=(b;q,p)rn(aq;q,p)rnθ(aq(r+1)k;p)θ(bq(r−1)k;p)θ(a;p)θ(b;p)×(a,1/b;q,p)k(qr,abqr;qr,p)k(abqrn/b,q−rn;qr,p)k(q1−rn,aqrn+1;q,p)kqk |
and
gn,k=θ(abq2rk;p)θ(ab;p)(aqn;q,p)rk(bq1−n;q,p)rk(ab,q−rn;qr,p)k(qr,abqrn+r;qr,p)kqrk. |
It is due to Warnaar [14,Eq.(3.5)], who used it to set up a series of summation and transformation formulas for terminating, balanced, very-well poised, elliptic hypergeometric series.
The work was supported by the National Natural Science Foundation of China (grant 11771330). The authors would like to thank the referees for their careful reading of the manuscript and for their constructive suggestions provided in the reports.
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